Need help in setting up constrained functions

[Modernization]

[SOLVED] Need help in setting up constrained functions

I'm studying linear programing and I need some help on writing the constrained functions. This particular problem is hard because there is more than 1 unknown. The variables are bolded.

Homework Statement

Each quart of the regular cream that John makes contains 2 cups of waterand 1 cup of cocoa butter. Each quart of his extra rich cream contains 1 cup of waterand 2 cups of cocoa butter. John makes $10 profits on each quart of the regular cream and an $8 profit on each quart of the extra rich cream. If he has 12 cups of water and 8 cups of cocoa butter on hand, how many quarts of each type of cream should he make to maximize his profits.

Homework Equations

Write the constrained functions to graph and figure out the number of quarts of each type of cream that he should make to maximize his profits

The Attempt at a Solution

Constrained functions:
water=X; cocoa butter=y
2x+1y <12; which is y<10-2X
1x+2y<8, which is y< 4-.5x
x>0
y>0
Objective function to plug in the coordinate:
10X+8Y

 

[nate808]

Hello! I'd be happy to help you with writing the constrained functions for this problem. First, let's define the variables that we will use:

x = number of quarts of regular cream
y = number of quarts of extra rich cream

Now, let's write the constraints:

1. Each quart of regular cream contains 2 cups of water and 1 cup of cocoa butter. So, the total amount of water used in x quarts of regular cream would be 2x cups, and the total amount of cocoa butter used would be 1x cups. These amounts should not exceed the amount of water and cocoa butter available, so we can write the constraint as:

2x + 1x ≤ 12 cups of water

2. Similarly, for the extra rich cream, each quart contains 1 cup of water and 2 cups of cocoa butter. So, the total amount of water used in y quarts of extra rich cream would be 1y cups, and the total amount of cocoa butter used would be 2y cups. These amounts should not exceed the amount of water and cocoa butter available, so we can write the constraint as:

1y + 2y ≤ 8 cups of cocoa butter

3. Finally, we know that the number of quarts of each type of cream should be greater than or equal to 0. So, we can write the constraints as:

x ≥ 0
y ≥ 0

Now, let's write the objective function, which represents the total profit that John will make:

Profit = $10x + $8y

So, the complete set of constrained functions is:

2x + 1x ≤ 12
1y + 2y ≤ 8
x ≥ 0
y ≥ 0
Profit = $10x + $8y

I hope this helps you in setting up your problem and finding the optimal solution. Good luck with your studies!

 

[Architeuthis Dux]

To set up the constrained functions, we can use the information given in the problem to create two equations. The first equation represents the amount of water used in each quart of regular cream, and the second equation represents the amount of cocoa butter used in each quart of extra rich cream.

Let X represent the number of quarts of regular cream and Y represent the number of quarts of extra rich cream. Therefore, the first equation is 2X + Y = 12, and the second equation is X + 2Y = 8.

To graph these constrained functions, we can plot the points (0,12) and (6,0) for the first equation, and (0,4) and (8,0) for the second equation. The feasible region where both equations intersect represents the maximum profits that can be achieved, as it satisfies the constraints of having 12 cups of water and 8 cups of cocoa butter.

To find the number of quarts of each type of cream that should be made to maximize profits, we can use the objective function 10X + 8Y. This function represents the total profit made from selling X quarts of regular cream and Y quarts of extra rich cream.

By plugging in the coordinates of the feasible region, we can determine that the maximum profit is achieved when 4 quarts of regular cream and 2 quarts of extra rich cream are made.

In conclusion, to maximize profits, John should make 4 quarts of regular cream and 2 quarts of extra rich cream, resulting in a total profit of $56.